Find a lambda-expressions F, K such that for all M, FM = F and KM = MK

Question

Find a lambda-expression F such that for all M, FM = F

Find a lambda-expression K such that for all M, KM = MK

My guess is to somehow use the combinator Y := \f. (\x.f(xx))(\x.f(xx)) so that YF = F(YF)

Answer 1

This Platform is not for doing home works ! Please avoid HW questions as much as possible. For using F = (\fx.ff)(\fx.ff) is one such use that leads to:

FM   ~=  (\fx.ff)(\fx.ff)M 
     **> (\fx.ff)(\fx.ff) 
     ~=  F

I think It's enough that u'll get the taste what you have to do for next. Hint for your solution is just think in terms of Y combinator and how it transforms the arguments.